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Validity, Unsatisfiability I The truth value of a propositional formula depends on truth CS311H: Discrete Mathematics assignments to variables I Example: p evaluates to true under under the assignment p = F Propositional Logic II and to false¬ under p = T I Some formulas evaluate to true for every assignment, e.g., p p Instructor: I¸sıl Dillig ∨ ¬ I Such formulas are called tautologies or valid formulas I Some formulas evaluate to false for every assignment, e.g., p p ∧ ¬ I Such formulas are called unsatisfiable formulas or contradictions Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 1/25 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 2/25 Interpretations Entailment I To make satisfability/validity precise, we’ll define I Under an interpretation, every propositional formula evaluates interpretation of formula to T or F Formula F + Interpretation I = Truth value I An interpretation I for a formula F is a mapping from each propositional variables in F to exactly one truth value I We write I = F if F evaluates to true under I | I : p true, q false, { 7→ 7→ ···} I Similarly, I = F if F evaluates to false under I . 6| I Each interpretation corresponds to one row in the truth table, n I Theorem: I = F if and only if I = F so 2 possible interpretations | 6| ¬ Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 3/25 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 4/25 Examples Another Example I Consider the formula F : p q p q ∧ → ¬ ∨ ¬ I Let I be the interpretation such that [p true, q false] 1 7→ 7→ I Let F1 and F2 be two propositional formulas I What does F evaluate to under I1? I Suppose F1 evaluates to true under interpretation I I Thus, I = F 1 | I What does F2 F1 evaluate to under I ? I Let I be the interpretation such that [p true, q true] ∧ ¬ 2 7→ 7→ I What does F evaluate to under I2? I Thus, I = F 2 6| Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 5/25 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 6/25 1 Satisfiability, Validity True/False Questions Are the following statements true or false? I F is satisfiable iff there exists interpretation I s.t. I = F | I If a formula is valid, then it is also satisfiable. I F is valid iff for all interpretations I , I = F | I If a formula is satisfiable, then its negation is unsatisfiable. I F is unsatisfiable iff for all interpretations I , I = F 6| I If F1 and F2 are satisfiable, then F1 F2 is also satisfiable. I F is contingent if it is satisfiable, but not valid. ∧ I If F and F are satisfiable, then F F is also satisfiable. 1 2 1 ∨ 2 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 7/25 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 8/25 Duality Between Validity and Unsatisfiability Proving Validity I Question: How can we prove that a propositional formula is a tautology? F is valid if and only if F is unsatisfiable ¬ I Exercise: Which formulas are tautologies? Prove your answer. 1. (p q) ( q p) I Proof: → ↔ ¬ → ¬ 2. (p q) p ∧ ∨ ¬ Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 9/25 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 10/25 Proving Satisfiability, Unsatisfiability, Contingency Exercise I Similarly, can prove satisfiability, unsatisfiability, contingency I Is (p q) (q p) valid, unsatisfiable, or contingent? using truth tables: → → → Prove your answer. I Satisfiable: There exists a row where formula evaluates to true I Unsatisfiable: In all rows, formula evaluates to false I Contingent: Exists a row where formula evaluates to true, and another row where it evaluates to false Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 11/25 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 12/25 2 Implication Example I Does p q imply p? Prove your answer. ∨ I Formula F implies F (written F F ) iff for all 1 2 1 ⇒ 2 interpretations I , I = F F | 1 → 2 F F iff F F is valid 1 ⇒ 2 1 → 2 I Caveat: F F is not a propositional logic formula; is 1 ⇒ 2 ⇒ not part of PL syntax! I Instead, F F is a semantic judgment, like satisfiability! 1 ⇒ 2 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 13/25 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 14/25 Equivalence Example I Two formulas F1 and F2 are equivalent if they have same I Prove that p q and p q are equivalent truth value for every interpretation, e.g., p p and p → ¬ ∨ ∨ I More precisely, formulas F1 and F2 are equivalent, written F F or F F , iff: 1 ≡ 2 1 ⇔ 2 F F iff F F is valid 1 ⇔ 2 1 ↔ 2 I , not part of PL syntax; they are semantic judgments! ≡ ⇔ Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 15/25 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 16/25 Important Equivalences Commutativity and Distributivity Laws I Commutative Laws: p q q p p q q p I Some important equivalences are useful to know! ∨ ≡ ∨ ∧ ≡ ∧ I Distributivity Law #1: (p (q r)) (p q) (p r) I Law of double negation: p p ∨ ∧ ≡ ∨ ∧ ∨ ¬¬ ≡ I Distributivity Law #2: (p (q r)) (p q) (p r) I Identity Laws: p T p p F p ∧ ∨ ≡ ∧ ∨ ∧ ∧ ≡ ∨ ≡ I Associativity Laws: p (q r) (p q) r I Domination Laws: p T T p F F ∨ ∨ ≡ ∨ ∨ ∨ ≡ ∧ ≡ p (q r) (p q) r ∧ ∧ ≡ ∧ ∧ I Idempotent Laws: p p p p p p ∨ ≡ ∧ ≡ I Absorption #1: p (p q) p ∧ ∨ ≡ I Negation Laws: p p F p p T ∧ ¬ ≡ ∨ ¬ ≡ I Absorption #2: p (p q) p ∨ ∧ ≡ Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 17/25 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 18/25 3 De Morgan’s Laws Why are These Equivalences Useful? I Let cs311 be the proposition ”John took CS311” and cs312be the proposition ”John took CS312” I Use known equivalences to prove that two formulas are equivalent by rewriting one formula to another I In simple English what does (cs311 cs312) mean? ¬ ∧ I Examples: Prove following formulas are equivalent: I DeMorgan’s law expresses exactly this equivalence! 1. (p ( p q)) and p q ¬ ∨ ¬ ∧ ¬ ∧ ¬ I De Morgan’s Law #1: (p q) ( p q) ¬ ∧ ≡ ¬ ∨ ¬ 2. (p q) and p q ¬ → ∧ ¬ I De Morgan’s Law #2: (p q) ( p q) ¬ ∨ ≡ ¬ ∧ ¬ Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 19/25 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 20/25 Formalizing English Arguments in Logic Example, cont I We can use logic to prove correctness of English arguments. ”If Joe drives fast, he gets a speeding ticket. Joe did not get a ticket. Therefore, he did not drive fast.”: ((f t) t) f I For example, consider the argument: → ∧ ¬ → ¬ I If Joe drives fast, he gets a speeding ticket. I How can we prove this argument is valid? I Joe did not get a ticket. I Can do this in two ways: 1. Use truth table to show formula is tautology I Therefore, Joe did not drive fast. 2. Use known equivalences to rewrite formula to true I Let f be the proposition ”Joe drives fast”, and t be the proposition ”Joe gets a ticket” I Let’s use equivalences I How do we encode this argument as a logical formula? Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 21/25 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 22/25 Another Example Example, cont. I Can also use to logic to prove an argument is not valid. I Suppose your friend George make the following argument: I If Jill carries an umbrella, it is raining. ”If Jill carries an umbrella, it is raining. Jill is not carrying an umbrella. Therefore it is not raining.”: ((u r) u) r I Jill is not carrying an umbrella. → ∧ ¬ → ¬ I How can we prove George’s argument is invalid? I Therefore it is not raining. I Let’s use logic to prove George’s argument doesn’t hold water. I Let u = ”Jill is carrying an umbrella”, and r = ”It is raining” I How do we encode this argument in logic? Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 23/25 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 24/25 4 Summary I A formula is valid if it is true for all interpretations. I A formula is satisfiable if it is true for at least one interpretation. I A formula is unsatisfiable if it is false for all interpretations. I A formula is contingent if it is true in at least one interpretation, and false in at least one interpretation. I Two formulas F and F are equivalent, written F F , if 1 2 1 ≡ 2 F F is valid 1 ↔ 2 Instructor: I¸sıl Dillig, CS311H: Discrete Mathematics Propositional Logic II 25/25 5.
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